By Chjan C Lim; Xueru Ding; Joseph Nebus

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**Extra resources for Vortex dynamics, statistical mechanics, and planetary atmospheres**

**Sample text**

9) Now, we will look at the rest-frame kinetic energy, given in terms of variables in the frame which is rotating at the same fixed angular velocity Ω of the planet. 12) S2 We can find the truncated, or lattice approximate, energy LN by putting into the above formula for rest-frame kinetic energy our approximation to February 24, 2009 14:26 World Scientific Book - 9in x 6in masterb Barotropic and Shallow-Water Models the relative vorticity ω. 15) following the rules of Lebesgue integration. 18) k=1 The interactions Jj,k are logarithmic in terms of the distance between points xj and xk and they are, therefore, long-range.

But — is there a better constraint possible, or is there a better reason than computational convenience that could be brought to it? 54) is fixed: it is the square of the L2 norm of the fixed planetary vorticity. 54) is proportional to the potentially varying net angular momentum of the atmosphere relative to the rotating frame. 54) is the relative enstrophy. In the Barotropic Vorticity Equation both the total enstrophy and relative enstrophy can — depending on the particular context of the problems — in principle be conserved in the coupled fluid-sphere model.

3 the L2 norm for the function cos (θ) and the spherical harmonic function ψ1,0 which is also the relative vorticity of solid-body rotation. 1. In this chapter we decompose the relative vorticity ω into its spectral components. One question which comes immediately to mind is in what space of functions should we consider ω to be? There are some physical properties of the relative vorticity which drive our choice. In particular we know the circulation and the enstrophy of the atmosphere are properties we want to measure.